Cz⁢(R)={(x,y,z)∈ℝ3∣∃r∈ℝ⁢ such that ⁢(x,y,r)∈R}.subscriptCzRconditional-setxyzsuperscriptℝ3rℝ such that xyrRC_{z}(R)=\{(x,y,z)\in\mathbb{R}^{3}\mid\exists r\in\mathbb{R}\mbox{ such that %
}(x,y,r)\in R\}.

Cz⁢(R)subscriptCzRC_{z}(R) is called the cylindrification of RRR with respect to the variablezzz.
It is easy to see that the characterization above permits us to generalize the notion of cylindrification to any subset of ℝℝ\mathbb{R}, with respect to any of the three variables x,y,zxyzx,y,z. We have in addition to (1) above the following properties:

where u,v∈{x,y,z}uvxyzu,v\in\{x,y,z\} and R,S⊆ℝ3RSsuperscriptℝ3R,S\subseteq\mathbb{R}^{3}.

Property (2) is obvious. To see Property (3), it is enough to assume u=zuzu=z (for the other cases follow similarly). First let (a,b,c)∈Cz⁢(R∩Cz⁢(S))abcsubscriptCzRsubscriptCzS(a,b,c)\in C_{z}(R\cap C_{z}(S)). Then there is an r∈ℝrℝr\in\mathbb{R} such that (a,b,r)∈RabrR(a,b,r)\in R and (a,b,r)∈Cz⁢(S)abrsubscriptCzS(a,b,r)\in C_{z}(S), which means there is an s∈ℝsℝs\in\mathbb{R} such that (a,b,s)∈SabsS(a,b,s)\in S. Since (a,b,r)∈RabrR(a,b,r)\in R, we have that (a,b,c)∈Cz⁢(R)abcsubscriptCzR(a,b,c)\in C_{z}(R), and since (a,b,s)∈SabsS(a,b,s)\in S, we have that (a,b,c)∈Cz⁢(S)abcsubscriptCzS(a,b,c)\in C_{z}(S) as well. This shows one inclusion. Now let (a,b,c)∈Cz⁢(R)∩Cz⁢(S)abcsubscriptCzRsubscriptCzS(a,b,c)\in C_{z}(R)\cap C_{z}(S), then there is an r∈ℝrℝr\in\mathbb{R} such that (a,b,r)∈RabrR(a,b,r)\in R. But (a,b,r)∈Cz⁢(S)abrsubscriptCzS(a,b,r)\in C_{z}(S) also, so (a,b,c)∈Cz⁢(R∩Cz⁢(S))abcsubscriptCzRsubscriptCzS(a,b,c)\in C_{z}(R\cap C_{z}(S)). To see Property (4), it is enough to assume u=xuxu=x and v=yvyv=y. Let (a,b,c)∈Cx⁢(Cy⁢(R))abcsubscriptCxsubscriptCyR(a,b,c)\in C_{x}(C_{y}(R)). Then there is an r∈ℝrℝr\in\mathbb{R} such that (r,b,c)∈Cy⁢(R)rbcsubscriptCyR(r,b,c)\in C_{y}(R), and so there is an s∈ℝsℝs\in\mathbb{R} such that (r,s,c)∈RrscR(r,s,c)\in R. This implies that (a,s,c)∈Cx⁢(R)ascsubscriptCxR(a,s,c)\in C_{x}(R), which implies that (a,b,c)∈Cy⁢(Cx⁢(R))abcsubscriptCysubscriptCxR(a,b,c)\in C_{y}(C_{x}(R)). So Cx⁢(Cy⁢(R))⊆Cy⁢(Cx⁢(R))subscriptCxsubscriptCyRsubscriptCysubscriptCxRC_{x}(C_{y}(R))\subseteq C_{y}(C_{x}(R)). The other inclusion then follows immediately.

with respect to xxx and yyy. This is just the plane whose projection onto the xxx-yyy plane is the linex=yxyx=y. We may define a total of nine possible diagonal sets Dv⁢wsubscriptDvwD_{{vw}} where v,w∈{x,y,z}vwxyzv,w\in\{x,y,z\}. However, there are in fact four distinct diagonal sets, since

where u,v∈{x,y,z}uvxyzu,v\in\{x,y,z\}. For any subset R⊆ℝ3Rsuperscriptℝ3R\subseteq\mathbb{R}^{3}, set Ru⁢v:=R∩Du⁢vassignsubscriptRuvRsubscriptDuvR_{{uv}}:=R\cap D_{{uv}}. For instance, Rx⁢y={(a,b,c)∈R∣a=b}subscriptRxyconditional-setabcRabR_{{xy}}=\{(a,b,c)\in R\mid a=b\}.

We may consider Cx,Cy,CzsubscriptCxsubscriptCysubscriptCzC_{x},C_{y},C_{z} as unaryoperations on ℝ3superscriptℝ3\mathbb{R}^{3}, and the diagonal sets as constants (nullary operations) on ℝ3superscriptℝ3\mathbb{R}^{3}. Two additional noteworthy properties are

To see Property (7), we may assume u=xuxu=x and v=yvyv=y. Suppose (a,b,c)∈Cx⁢(Rx⁢y)∩Cx⁢(Rx⁢y′)abcsubscriptCxsubscriptRxysubscriptCxsubscriptsuperscriptRnormal-′xy(a,b,c)\in C_{x}(R_{{xy}})\cap C_{x}(R^{{\prime}}_{{xy}}). Then there is r∈ℝrℝr\in\mathbb{R} such that (r,b,c)∈Rx⁢yrbcsubscriptRxy(r,b,c)\in R_{{xy}}, which implies that r=brbr=b, or that (b,b,c)∈RbbcR(b,b,c)\in R. On the other hand, there is s∈ℝsℝs\in\mathbb{R} such that (s,b,c)∈Rx⁢y′sbcsubscriptsuperscriptRnormal-′xy(s,b,c)\in R^{{\prime}}_{{xy}}, which implies s=bsbs=b, or that (b,b,c)∈R′bbcsuperscriptRnormal-′(b,b,c)\in R^{{\prime}}, a contradiction. To see Property (8), we may assume u=x,v=w,w=zformulae-sequenceuxformulae-sequencevwwzu=x,v=w,w=z. If (a,b,c)∈Cx⁢(Dx⁢y∩Dx⁢z)abcsubscriptCxsubscriptDxysubscriptDxz(a,b,c)\in C_{x}(D_{{xy}}\cap D_{{xz}}), then there is r∈ℝrℝr\in\mathbb{R} such that (r,b,c)∈Dx⁢y∩Dx⁢zrbcsubscriptDxysubscriptDxz(r,b,c)\in D_{{xy}}\cap D_{{xz}}. So r=brbr=b and r=crcr=c. Therefore, (a,b,c)=(a,r,r)∈Dy⁢zabcarrsubscriptDyz(a,b,c)=(a,r,r)\in D_{{yz}}. On the other hand, for any (a,r,r)∈Dy⁢zarrsubscriptDyz(a,r,r)\in D_{{yz}}, (r,r,r)∈Dx⁢y∩Dx⁢zrrrsubscriptDxysubscriptDxz(r,r,r)\in D_{{xy}}\cap D_{{xz}}, and so (a,r,r)∈Cx⁢(Dx⁢y∩Dx⁢z)arrsubscriptCxsubscriptDxysubscriptDxz(a,r,r)\in C_{x}(D_{{xy}}\cap D_{{xz}}) as well.

Finally, we note that a subset of ℝ3superscriptℝ3\mathbb{R}^{3} is just a ternary relation on ℝℝ\mathbb{R}, and the collection of all ternary relations on RRR is just P⁢(ℝ3)Psuperscriptℝ3P(\mathbb{R}^{3}).

Proposition 1.

P⁢(ℝ3)Psuperscriptℝ3P(\mathbb{R}^{3}) is a Boolean algebra with the usual set-theoretic operations, and together with cylindrification operators and the diagonal sets, on the set V={x,y,z}VxyzV=\{x,y,z\}, is a cylindric algebra.

Proof.

Write A=P⁢(ℝ3)APsuperscriptℝ3A=P(\mathbb{R}^{3}). It is easy to see that AAA is a Boolean algebra with operations ∪,∩,′,∅fragmentsnormal-,superscriptnormal-,normal-′normal-,\cup,\cap,^{{\prime}},\varnothing. Next define ∃:V→AAnormal-:normal-→VsuperscriptAA\exists:V\to A^{A} by ∃v:=CvassignvsubscriptCv\exists v:=C_{v} where v∈{x,y,z}vxyzv\in\{x,y,z\}, and d:V×V→Anormal-:dnormal-→VVAd:V\times V\to A by dx⁢y:=Dx⁢yassignsubscriptdxysubscriptDxyd_{{xy}}:=D_{{xy}}. Then Properties (1), (2), and (3) show that (A,∃v)Asubscriptv(A,\exists_{v}) is a monadic algebra, and Properties (4), (5), (7), and (8) show that (A,V,∃,d)AVd(A,V,\exists,d) is cylindric.
∎

Example 2 (Cylindric Set Algebras).

Example 1 above may be generalized. Let A,VAVA,V be sets, and set B=P⁢(AV)BPsuperscriptAVB=P(A^{V}). For any subset R⊆BRBR\subseteq B and any x,y∈VxyVx,y\in V, define the cylindrification of RRR by

Cx⁢(R):={p∈AV∣∃r∈R⁢ such that ⁢r⁢(y)=p⁢(y)⁢ for any ⁢y≠x},assignsubscriptCxRconditional-setpsuperscriptAVrR such that rypy for any yxC_{x}(R):=\{p\in A^{V}\mid\exists r\in R\mbox{ such that }r(y)=p(y)\mbox{ for %
any }y\neq x\},